I was thinking about adjacency matrices, when I realised that an adjacency matrix is nilpotent if and only if there is no cycle in the graph.
So, I started wondering: if a graph has an adjacency matrix that is invertible, what does it mean for the graph?
I usually define a graph the following way (but if the is some proprieties with another definition, I'm also interested): a graph $G=(V,E)$ is couple of set where $V = [\![1;n]\!]$ and $E \subset \{ (i,j) \in V^2, i\neq j$}. The adjacency matrix $M=(M_{ij})$ is defined such as $M_{ij} = 1$ if $(i,j) \in V$, $0$ otherwise.
Here I'm talking about undirected graphs.